Final answer:
The number which, when combined with 294 yields an LCM of 4,410 is 15. This is determined by dividing 4,410 by 294 after finding that 4410 is a multiple of 294.
Step-by-step explanation:
The student is asking us to find the number that, when combined with 294, will have a Least Common Multiple (LCM) of 4,410. First, we'll want to factor the LCM (4,410) and the known number (294) into their prime factors. To find the other number that has an LCM of 4410 with 294, we divide 4410 by the greatest common divisor (GCD) of 294 and 4410.
Let's find the prime factors of 294: 294 is 2 × 3 × 7 × 7. Now, finding the LCM means that we're looking for a number that includes these factors and possibly more. If the LCM of 294 and another number is 4410, then 4410 must include the prime factors of 294 and any additional factors of the other number that are not already included in 294.
In this case, the GCD of 294 and 4410 is 294, as 4410 is a multiple of 294. Thus, to find the other number we would divide 4410 by 294. Performing this division gives us 15, which means the other number has to have the prime factors that, when multiplied with those of 294, will give 4410 without introducing any further common factors.
Therefore, the other number that has an LCM of 4410 with 294 is indeed 15, as 15 has the prime factor of 3 and 5 which, when multiplied by the prime factors of 294, give us the LCM of 4410.